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Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$is some integer. We prove that all such ideals are obtained by a trimmingprocess introduced by Christensen, Veliche, and Weyman. We also construct ageneral resolution for all such ideals which is minimal in sufficiently genericcases. Using this resolution, we can give bounds on the minimal number ofgenerators $\mu(I)$ of $I$ depending only on $s$; moreover, we show thesebounds are sharp by constructing ideals attaining the upper and lower boundsfor all $s\geq 3$. Finally, we study the Tor-algebra structure of $R/I$. It isshown that these rings have Tor algebra class $G(r)$ for $s \leq r \leq 2s-1$.Furthermore, we produce ideals $I$ for all $s \geq 3$ and all $r$ with $s \leqr \leq 2s-1$ such that $\textrm{Soc} (R/I ) = k(-s) \oplus k(-2s+1)$ and $R/I$has Tor-algebra class $G(r)$, partially answering a question of realizabilityposed by Avramov.

Structure Theory for a Class of Grade 3 Homogeneous Ideals Defining Type 2 Compressed Rings